Numerical Solution of Convection-Diffusion problems · PDF fileNumerical Solution of Convection-Diffusion problems ... Numerical solution of convection-diffusion problems, ... Equations

Finite Volume methods for steady problems

Numerical Solution of Convection-Diffusion problems

Remo Minero

June 1, 2005Finite Volume methods for steady problems2

Seminarhttp://www.win.tue.nl/casa/meetings/seminar/index.html

May 18: Numerical solution of convection-diffusion problems, an introduction

May 25: Numerical solution of convection-diffusion problems: difference schemes for steady problems

June 29 July 6


Outline

Main idea of finite volumeconservationdifferent approaches

Numerical discretizationquadrature rulesinterpolation schemes

Different grid types


The main idea

Ω+Φ∇Γ⋅∇=Φ∇⋅ρ ins)(vConvection-diffusionequation

ρv field is divergence freeApply Gauss’ theorem

Want a numerical method that satisfies a discrete equivalent of

Ω⊆+⋅Φ∇Γ=⋅Φρ ∫∫∫ VanyfordVsdSdSVSS

nnv


Example (time-dep. problem)

Φ is the concentration of a passive tracer transported by a velocity field v in a closed domain.

Because of the walls: v · n = 0 ∇Φ · n = 0

Analytical solution:

Numerical solution: a discrete equivalent of

Ω⊆⋅Φ∇Γ=⋅Φρ+∂Φ∂

∫∫∫ VanyfordSdSdVt SSV

nnv

∫Ω

=Φ constdV


Different finite volume schemesControl volumes

cell vertex

cell center

vertexcenteredcell

edge


Numerical discretization

Ω⊆+⋅Φ∇Γ=⋅Φρ ∫∫∫ VanyfordVsdSdSVSS

nnv

1. Quadrature rule

2. Interpolationscheme

(Cell centered approach)


Three-dimensional case


Fluxes

Flux f(Φ)

Integrated flux F(Φ)

Equation

∫∫∫ ⋅Φ∇Γ−⋅Φρ=⋅=SSS

dSdSdSF nnvnf

Φ∇Γ−Φρ= vf

∫∫ =⋅VS

dvsdSnf

convective flux fc

diffusive flux fd


Approximation of surface integrals

Midpoint rule (2nd order)

Trapezoidal rule (2nd order)

Simpson’s rule (4th order)

eeS

e SfdSFe

≈⋅= ∫ nf

)ff(21SdSF senee

Se

e

+≈⋅= ∫ nf

)ff4f(61SdSF seenee

Se

e

++≈⋅= ∫ nf


Approximation of volume integrals

2nd order formula

4th order formula (uniform Cartesian grid)

yxsdVs Pv

∆∆≈∫

)ssss

s4s4s4s4s16(36yxdVs

nwneswse

wensPv

+++

+++++∆∆≈∫


Interpolation schemes: UDS

Upwind differencing scheme (UDS)

Never oscillatory solutionsArtificial diffusion:

<⋅Φ>⋅Φ

=Φ0)(if0)(if

eE

ePe nv

nv

e

ee

e

nume

de x2

xvx

f

∂Φ∂∆ρ=

∂Φ∂Γ=

1st order accuracy


Interpolation schemes: CDS

Centered diff. scheme (CDS)

More accurate than UDSMay produce oscillatory solutions

2nd order accuracy

PE

eEP

PE

PeEe xx

xxxxxx

−−Φ+

−−Φ≈Φ

PE

PE

e xxx −Φ−Φ≈

∂Φ∂

fc

fd


High order interpolation schemes

QUICK (Quadratic Upwind Interpolation for Convective Kinematics)

4th order CDS

WEPe 81

83

86 Φ−Φ+Φ≈Φ ( )

P3

33

xx

483

∂Φ∂∆−

( )EEWEPe 332727481 Φ−Φ−Φ+Φ≈Φ

( )EEWPEe

2727x24

1x

Φ−Φ+Φ−Φ∆

≈

∂Φ∂


Remark on different schemes convective diffusive

cell vertex

cell center

vertexcenteredcell

edge

diffusive convective


Linear system and boundary conditions

Equations for ΦP form a linear system

System is closed by boundary conditionse.g. One side differencesfor diffusive fluxes

Example:2D, cell centered, midpoint rule + 2nd CDS | 1st UDS:pentadiagonal system


Deferred correction

High order schemes Large computational molecule2D, Simpson rule + 4th order CDS: each flux depends on 15 nodal values

Large computational molecule Expensive solution of linear system

Idea: combine low and high order approximationsHigh order approximation are only computed explicitly

( )oldLOWe

HIGHe

LOWee FFFF −+=


Example

Velocity field:v=(vx,vy) = (x,-y)

Density ρ = 1

Test UDS and CDS with midpoint rule and cell centered


Isolines of Φ for different Γ

0.05

0.15

0.25

0.35

0.45

0.550.650.750.85

0.95

0.05

0.15

Γ = 10-3Γ = 10-2


Convergence Q = Integrated flux throughthe west side of the domain

slope 2

slope 1


Different grid types

Tensor product grid

By construction, in neighboring control volumes influxes and outfluxes are balanced

sum of discrete conservation laws

Γ = 10-3


Composite grid

Balance of fluxes across the interface coarse-fine grid is not guaranteed

In Local Defect Correction (LDC)

iterative improvement between coarse and fine grid solutionin the limit: balance of fluxes everywhere


Time-dependent problem

Standard gridssum of conservationlaws also in time

Composite grid with different rates for time integrationLDC: balance preserved

ttntn-1

ttntn-1


Conclusions

The main ideas behind the finite volume methods were introduced

Schemes for quadrature and interpolation were discussed

Some issues about conservation on different grid types were addressed

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Numerical Solution of Convection-Diffusion problems · PDF fileNumerical Solution of Convection-Diffusion problems ... Numerical solution of convection-diffusion problems, ... Equations